Abstract

A unified view of the area of sparse signal processing is presented in tutorial form
by bringing together various fields in which the property of sparsity has been successfully
exploited. For each of these fields, various algorithms and techniques, which have
been developed to leverage sparsity, are described succinctly. The common potential
benefits of significant reduction in sampling rate and processing manipulations through
sparse signal processing are revealed. The key application domains of sparse signal
processing are sampling, coding, spectral estimation, array processing, component
analysis, and multipath channel estimation. In terms of the sampling process and reconstruction
algorithms, linkages are made with random sampling, compressed sensing, and rate of
innovation. The redundancy introduced by channel coding in finite and real Galois
fields is then related to over-sampling with similar reconstruction algorithms. The
error locator polynomial (ELP) and iterative methods are shown to work quite effectively
for both sampling and coding applications. The methods of Prony, Pisarenko, and MUltiple
SIgnal Classification (MUSIC) are next shown to be targeted at analyzing signals with
sparse frequency domain representations. Specifically, the relations of the approach
of Prony to an annihilating filter in rate of innovation and ELP in coding are emphasized;
the Pisarenko and MUSIC methods are further improvements of the Prony method under
noisy environments. The iterative methods developed for sampling and coding applications
are shown to be powerful tools in spectral estimation. Such narrowband spectral estimation
is then related to multi-source location and direction of arrival estimation in array
processing. Sparsity in unobservable source signals is also shown to facilitate source
separation in sparse component analysis; the algorithms developed in this area such
as linear programming and matching pursuit are also widely used in compressed sensing.
Finally, the multipath channel estimation problem is shown to have a sparse formulation;
algorithms similar to sampling and coding are used to estimate typical multicarrier
communication channels.